Advertisements
Advertisements
प्रश्न
Find the volume of the largest cylinder that can be inscribed in a sphere of radius ‘r’ cm.
Advertisements
उत्तर
Let R be the radius and h be the height of the cylinder which is inscribed in a sphere of radius r cm.
Then from the figure,
`"R"^2 + (h/2)^2` = r2
∴ R2 = `r^2 - h^2/(4)` ...(1)
Let V be the volume of the cylinder.
Then V = πR2h
= `pi(r^2 - h^2/(4))h` ...[By (1)]
= `pi(r^2 - h^3/(4))`
∴ `"dV"/"dh" = pid/"dh"(r^2h - h^3/(4))`
= `pi(r^2 xx 1 - 1/4 xx 3h^2)`
= `pi(r^2 - 3/4h^2)`
and
`(d^2V)/("dh"^2) = pid/"dh"(r^2 - 3/4h^2)`
= `pi(0 - 3/4 xx 2h)`
= `-(3)/(2)pih`
Now, `"dV"/"dh" = 0 "gives", pi(r^2 - 3/4h^2)` = 0
∴ `r^2 - 3/4h^2` = 0
∴ `(3)/(4)h^2` = r2
∴ h2 = `(4r^2)/(3)`
∴ h = `(2r)/sqrt(3)` ...[∵ h > 0]
and
`((d^2V)/(dh^2))_("at" h = (2r)/sqrt(3)`
= `-(3)/(2)pi xx (2r)/sqrt(3) < 0`
∴ V is maximum at h = `(2r)/sqrt(3)`
If h = `(2r)/sqrt(3)`, then from (1)
R2 = `r^2 - (1)/(4) xx (4r^2)/(3) = (2r^2)/(3)`
∴ volumeof the largest cylinder
= `pi xx (2r^2)/(3) xx (2r)/sqrt(3) = (4pir^3)/(3sqrt(3)`cu cm.
Hence, the volume of the largest cylinder inscribed in a sphere of radius 'r' cm = `(4pir^3)/(3sqrt(3)`cu cm.
APPEARS IN
संबंधित प्रश्न
If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).
Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.` Also, find the maximum volume.
An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.
Find the maximum and minimum value, if any, of the following function given by f(x) = 9x2 + 12x + 2
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
g(x) = x3 − 3x
Prove that the following function do not have maxima or minima:
f(x) = ex
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
At what points in the interval [0, 2π], does the function sin 2x attain its maximum value?
Find the maximum and minimum values of x + sin 2x on [0, 2π].
Find two positive numbers whose sum is 16 and the sum of whose cubes is minimum.
Of all the closed cylindrical cans (right circular), of a given volume of 100 cubic centimetres, find the dimensions of the can which has the minimum surface area?
A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum?
Show that the right circular cone of least curved surface and given volume has an altitude equal to `sqrt2` time the radius of the base.
Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is `tan^(-1) sqrt(2)`
The maximum value of `[x(x −1) +1]^(1/3)` , 0 ≤ x ≤ 1 is ______.
Find the points at which the function f given by f (x) = (x – 2)4 (x + 1)3 has
- local maxima
- local minima
- point of inflexion
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of material will be least when the depth of the tank is half of its width. If the cost is to be borne by nearby settled lower-income families, for whom water will be provided, what kind of value is hidden in this question?
Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .
Divide the number 20 into two parts such that sum of their squares is minimum.
A ball is thrown in the air. Its height at any time t is given by h = 3 + 14t – 5t2. Find the maximum height it can reach.
A box with a square base is to have an open top. The surface area of the box is 192 sq cm. What should be its dimensions in order that the volume is largest?
Solve the following : An open box with a square base is to be made out of given quantity of sheet of area a2. Show that the maximum volume of the box is `a^3/(6sqrt(3)`.
Solve the following : Show that the height of a right circular cylinder of greatest volume that can be inscribed in a right circular cone is one-third of that of the cone.
A metal wire of 36 cm length is bent to form a rectangle. Find its dimensions when its area is maximum.
The function f(x) = x log x is minimum at x = ______.
The function y = 1 + sin x is maximum, when x = ______
The minimum value of the function f(x) = 13 - 14x + 9x2 is ______
Let f have second derivative at c such that f′(c) = 0 and f"(c) > 0, then c is a point of ______.
The smallest value of the polynomial x3 – 18x2 + 96x in [0, 9] is ______.
The function f(x) = 2x3 – 3x2 – 12x + 4, has ______.
Find all the points of local maxima and local minima of the function f(x) = (x - 1)3 (x + 1)2
Find the maximum profit that a company can make, if the profit function is given by P(x) = 41 + 24x – 18x2.
If y = x3 + x2 + x + 1, then y ____________.
The function f(x) = x5 - 5x4 + 5x3 - 1 has ____________.
Find the volume of the largest cylinder that can be inscribed in a sphere of radius r cm.
The area of a right-angled triangle of the given hypotenuse is maximum when the triangle is ____________.
The combined resistance R of two resistors R1 and R2 (R1, R2 > 0) is given by `1/"R" = 1/"R"_1 + 1/"R"_2`. If R1 + R2 = C (a constant), then maximum resistance R is obtained if ____________.
Range of projectile will be maximum when angle of projectile is
If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to ______.
If y = alog|x| + bx2 + x has its extremum values at x = –1 and x = 2, then ______.
The minimum value of 2sinx + 2cosx is ______.
Read the following passage:
Engine displacement is the measure of the cylinder volume swept by all the pistons of a piston engine. The piston moves inside the cylinder bore.
|
Based on the above information, answer the following questions:
- If the radius of cylinder is r cm and height is h cm, then write the volume V of cylinder in terms of radius r. (1)
- Find `(dV)/(dr)`. (1)
- (a) Find the radius of cylinder when its volume is maximum. (2)
OR
(b) For maximum volume, h > r. State true or false and justify. (2)
A metal wire of 36 cm long is bent to form a rectangle. Find its dimensions when its area is maximum.
If Mr. Rane order x chairs at the price p = (2x2 - 12x - 192) per chair. How many chairs should he order so that the cost of deal is minimum?
Solution: Let Mr. Rane order x chairs.
Then the total price of x chairs = p·x = (2x2 - 12x- 192)x
= 2x3 - 12x2 - 192x
Let f(x) = 2x3 - 12x2 - 192x
∴ f'(x) = `square` and f''(x) = `square`
f'(x ) = 0 gives x = `square` and f''(8) = `square` > 0
∴ f is minimum when x = 8
Hence, Mr. Rane should order 8 chairs for minimum cost of deal.
Find the point on the curve y2 = 4x, which is nearest to the point (2, 1).
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
f(x) `= x sqrt(1 - x), 0 < x < 1`
